Geometric interpretation of grade n elements in a real exterior algebra for n = 0 (signed point), 1 (directed line segment, or vector), 2 (oriented plane element), 3 (oriented volume). The exterior product of n vectors can be visualized as any n-dimensional shape (e.g. n-parallelotope, n-ellipsoid); with magnitude (hypervolume), and orientation defined by that on its (n − 1)-dimensional boundary and on which side the interior is.[1][2]

In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs. The exterior product of two vectors u and v, denoted by u ∧ v, is called a bivector and lives in a space called the exterior square, a geometrical vector space that differs from the original space of vectors. The magnitude[3] of u ∧ v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. Also like the cross product, the exterior product is anticommutative, meaning that u ∧ v = −(v ∧ u) for all vectors u and v. One way to visualize a bivector is as a family of parallelograms all lying in the same plane, having the same area, and with the same orientation of their boundaries—a choice of clockwise or counterclockwise.

When regarded in this manner the exterior product of two vectors is called a 2-blade. More generally, the exterior product of any number k of vectors can be defined and is sometimes called a k-blade. It lives in a geometrical space known as the k-th exterior power. The magnitude of the resulting k-blade is the volume of the k-dimensional parallelotope whose sides are the given vectors, just as the magnitude of the scalar triple product of vectors in three dimensions gives the volume of the parallelepiped spanned by those vectors.

The exterior algebra, or Grassmann algebra after Hermann Grassmann,[4] is the algebraic system whose product is the exterior product. The exterior algebra provides an algebraic setting in which to answer geometric questions. For instance, whereas blades have a concrete geometrical interpretation, objects in the exterior algebra can be manipulated according to a set of unambiguous rules. The exterior algebra contains objects that are not just k-blades, but sums of k-blades; such a sum is called a k-vector.[5] The k-blades, because they are simple products of vectors, are called the simple elements of the algebra. The rank of any k-vector is defined to be the smallest number of simple elements of which it is a sum. The exterior product extends to the full exterior algebra, so that it makes sense to multiply any two elements of the algebra. Equipped with this product, the exterior algebra is an associative algebra, which means that α ∧ (β ∧ γ) = (α ∧ β) ∧ γ for any elements α, β, γ. The k-vectors have degree k, meaning that they are sums of products of k vectors. When elements of different degrees are multiplied, the degrees add like multiplication of polynomials. This means that the exterior algebra is a graded algebra.

The definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as vector fields or functions. In full generality, the exterior algebra can be defined for modules over a commutative ring, and for other structures of interest in abstract algebra. It is one of these more general constructions where the exterior algebra finds one of its most important applications, where it appears as the algebra of differential forms that is fundamental in areas that use differential geometry. Differential forms are mathematical objects that represent infinitesimal areas of infinitesimal parallelograms (and higher-dimensional bodies), and so can be integrated over surfaces and higher dimensional manifolds in a way that generalizes the line integrals from calculus. The exterior algebra also has many algebraic properties that make it a convenient tool in algebra itself. The association of the exterior algebra to a vector space is a type of functor on vector spaces, which means that it is compatible in a certain way with linear transformations of vector spaces. The exterior algebra is one example of a bialgebra, meaning that its dual space also possesses a product, and this dual product is compatible with the exterior product. This dual algebra is precisely the algebra of alternating multilinear forms on V, and the pairing between the exterior algebra and its dual is given by the interior product.

are a pair of given vectors in R2, written in components. There is a unique parallelogram having v and w as two of its sides. The area of this parallelogram is given by the standard determinant formula:

Consider now the exterior product of v and w:

where the first step uses the distributive law for the exterior product, and the last uses the fact that the exterior product is alternating, and in particular e2 ∧ e1 = −e1 ∧ e2. Note that the coefficient in this last expression is precisely the determinant of the matrix [vw]. The fact that this may be positive or negative has the intuitive meaning that v and w may be oriented in a counterclockwise or clockwise sense as the vertices of the parallelogram they define. Such an area is called the signed area of the parallelogram: the absolute value of the signed area is the ordinary area, and the sign determines its orientation.

The fact that this coefficient is the signed area is not an accident. In fact, it is relatively easy to see that the exterior product should be related to the signed area if one tries to axiomatize this area as an algebraic construct. In detail, if A(v, w) denotes the signed area of the parallelogram determined by the pair of vectors v and w, then A must satisfy the following properties:

A(jv, kw) = j k A(v, w) for any real numbers j and k, since rescaling either of the sides rescales the area by the same amount (and reversing the direction of one of the sides reverses the orientation of the parallelogram).

A(w,v) = −A(v,w), since interchanging the roles of v and w reverses the orientation of the parallelogram.

A(v + jw,w) = A(v,w), for real j, since adding a multiple of w to v affects neither the base nor the height of the parallelogram and consequently preserves its area.

A(e1, e2) = 1, since the area of the unit square is one.

With the exception of the last property, the exterior product satisfies the same formal properties as the area. In a certain sense, the exterior product generalizes the final property by allowing the area of a parallelogram to be compared to that of any "standard" chosen parallelogram (here, the one with sides e1 and e2). In other words, the exterior product in two dimensions provides a basis-independent formulation of area.[6]

The cross product (blue vector) in relation to the exterior product (light blue parallelogram). The length of the cross product is to the length of the parallel unit vector (red) as the size of the exterior product is to the size of the reference parallelogram (light red).

For vectors in R3, the exterior algebra is closely related to the cross product and triple product. Using the standard basis {e1, e2, e3}, the exterior product of a pair of vectors

and

is

where {e1 ∧ e2, e3 ∧ e1, e2 ∧ e3} is the basis for the three-dimensional space Λ2(R3). The coefficients above are the same as those in the usual definition of the cross product of vectors in three dimensions, the only difference being that the exterior product is not an ordinary vector, but instead is a 2-vector.

Bringing in a third vector

the exterior product of three vectors is

where e1 ∧ e2 ∧ e3 is the basis vector for the one-dimensional space Λ3(R3). The scalar coefficient is the triple product of the three vectors.

The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations. The cross product u × v can be interpreted as a vector which is perpendicular to both u and v and whose magnitude is equal to the area of the parallelogram determined by the two vectors. It can also be interpreted as the vector consisting of the minors of the matrix with columns u and v. The triple product of u, v, and w is geometrically a (signed) volume. Algebraically, it is the determinant of the matrix with columns u, v, and w. The exterior product in three dimensions allows for similar interpretations. In fact, in the presence of a positively oriented orthonormal basis, the exterior product generalizes these notions to higher dimensions.

The exterior algebra Λ(V) over a vector space V over a fieldK is defined as the quotient algebra of the tensor algebra by the two-sided idealI generated by all elements of the form x ⊗ x such that x ∈ V.[7] Symbolically,

The exterior product ∧ of two elements of Λ(V) is defined by

where the mod I means we do the tensor product in the usual way and then declare every element of the tensor that is in the ideal to be zero.

If α ∈ Λk(V), then α is said to be a k-vector. If, furthermore, α can be expressed as an exterior product of k elements of V, then α is said to be decomposable. Although decomposable k-vectors span Λk(V), not every element of Λk(V) is decomposable. For example, in R4, the following 2-vector is not decomposable:

If the dimension of V is n and {e1,...,en} is a basis of V, then the set

is a basis for Λk(V). The reason is the following: given any exterior product of the form

then every vector vj can be written as a linear combination of the basis vectors ei; using the bilinearity of the exterior product, this can be expanded to a linear combination of exterior products of those basis vectors. Any exterior product in which the same basis vector appears more than once is zero; any exterior product in which the basis vectors do not appear in the proper order can be reordered, changing the sign whenever two basis vectors change places. In general, the resulting coefficients of the basis k-vectors can be computed as the minors of the matrix that describes the vectors vj in terms of the basis ei.

If α ∈ Λk(V), then it is possible to express α as a linear combination of decomposable k-vectors:

where each α(i) is decomposable, say

The rank of the k-vector α is the minimal number of decomposable k-vectors in such an expansion of α. This is similar to the notion of tensor rank.

Rank is particularly important in the study of 2-vectors (Sternberg 1974, §III.6) (Bryant et al. 1991). The rank of a 2-vector α can be identified with half the rank of the matrix of coefficients of α in a basis. Thus if ei is a basis for V, then α can be expressed uniquely as

where aij = −aji (the matrix of coefficients is skew-symmetric). The rank of the matrix aij is therefore even, and is twice the rank of the form α.

The exterior product of a k-vector with a p-vector is a (k+p)-vector, once again invoking bilinearity. As a consequence, the direct sum decomposition of the preceding section

gives the exterior algebra the additional structure of a graded algebra. Symbolically,

Moreover, the exterior product is graded anticommutative, meaning that if α ∈ Λk(V) and β ∈ Λp(V), then

In addition to studying the graded structure on the exterior algebra, Bourbaki (1989) studies additional graded structures on exterior algebras, such as those on the exterior algebra of a graded module (a module that already carries its own gradation).

Let V be a vector space over the field K. Informally, multiplication in Λ(V) is performed by manipulating symbols and imposing a distributive law, an associative law, and using the identity v ∧ v = 0 for v ∈ V. Formally, Λ(V) is the "most general" algebra in which these rules hold for the multiplication, in the sense that any unital associative K-algebra containing V with alternating multiplication on V must contain a homomorphic image of Λ(V). In other words, the exterior algebra has the following universal property:[10]

Given any unital associative K-algebra A and any K-linear mapj : V → A such that j(v)j(v) = 0 for every v in V, then there exists precisely one unital algebra homomorphismf : Λ(V) → A such that j(v) = f(i(v)) for all v in V.

To construct the most general algebra that contains V and whose multiplication is alternating on V, it is natural to start with the most general algebra that contains V, the tensor algebraT(V), and then enforce the alternating property by taking a suitable quotient. We thus take the two-sided idealI in T(V) generated by all elements of the form v⊗v for v in V, and define Λ(V) as the quotient

(and use ∧ as the symbol for multiplication in Λ(V)). It is then straightforward to show that Λ(V) contains V and satisfies the above universal property.

As a consequence of this construction, the operation of assigning to a vector space V its exterior algebra Λ(V) is a functor from the category of vector spaces to the category of algebras.

Rather than defining Λ(V) first and then identifying the exterior powers Λk(V) as certain subspaces, one may alternatively define the spaces Λk(V) first and then combine them to form the algebra Λ(V). This approach is often used in differential geometry and is described in the next section.

Given a commutative ringR and an R-moduleM, we can define the exterior algebra Λ(M) just as above, as a suitable quotient of the tensor algebra T(M). It will satisfy the analogous universal property. Many of the properties of Λ(M) also require that M be a projective module. Where finite dimensionality is used, the properties further require that M be finitely generated and projective. Generalizations to the most common situations can be found in (Bourbaki 1989).

Exterior algebras of vector bundles are frequently considered in geometry and topology. There are no essential differences between the algebraic properties of the exterior algebra of finite-dimensional vector bundles and those of the exterior algebra of finitely generated projective modules, by the Serre–Swan theorem. More general exterior algebras can be defined for sheaves of modules.

which associates to k vectors from V their exterior product, i.e. their corresponding k-vector, is also alternating. In fact, this map is the "most general" alternating operator defined on Vk: given any other alternating operator f : Vk → X, there exists a unique linear mapφ : Λk(V) → X with f = φ ∘ w. This universal property characterizes the space Λk(V) and can serve as its definition.

The above discussion specializes to the case when X = K, the base field. In this case an alternating multilinear function

is called an alternating multilinear form. The set of all alternating multilinear forms is a vector space, as the sum of two such maps, or the product of such a map with a scalar, is again alternating. By the universal property of the exterior power, the space of alternating forms of degree k on V is naturally isomorphic with the dual vector space (ΛkV)∗. If V is finite-dimensional, then the latter is naturally isomorphic to Λk(V∗). In particular, the dimension of the space of anti-symmetric maps from Vk to K is the binomial coefficientn choose k.

Under this identification, the exterior product takes a concrete form: it produces a new anti-symmetric map from two given ones. Suppose ω : Vk → K and η : Vm → K are two anti-symmetric maps. As in the case of tensor products of multilinear maps, the number of variables of their exterior product is the sum of the numbers of their variables. It is defined as follows:[13]

where the alternation Alt of a multilinear map is defined to be the signed average of the values over all the permutations of its variables:

This definition of the exterior product is well-defined even if the fieldK has finite characteristic, if one considers an equivalent version of the above that does not use factorials or any constants:

In formal terms, there is a correspondence between the graded dual of the graded algebra Λ(V) and alternating multilinear forms on V. The exterior product of multilinear forms defined above is dual to a coproduct defined on Λ(V), giving the structure of a coalgebra.

The coproduct is a linear function Δ : Λ(V) → Λ(V) ⊗ Λ(V) given on decomposable elements by

For example,

This extends by linearity to an operation defined on the whole exterior algebra. In terms of the coproduct, the exterior product on the dual space is just the graded dual of the coproduct:

where the tensor product on the right-hand side is of multilinear linear maps (extended by zero on elements of incompatible homogeneous degree: more precisely, α∧β = ε ∘ (α⊗β) ∘ Δ, where ε is the counit, as defined presently).

The counit is the homomorphism ε : Λ(V) → K which returns the 0-graded component of its argument. The coproduct and counit, along with the exterior product, define the structure of a bialgebra on the exterior algebra.

With an antipode defined on homogeneous elements by S(x) = (−1)deg xx, the exterior algebra is furthermore a Hopf algebra.[14]

Suppose that V is finite-dimensional. If V* denotes the dual space to the vector space V, then for each α ∈ V*, it is possible to define an antiderivation on the algebra Λ(V),

This derivation is called the interior product with α, or sometimes the insertion operator, or contraction by α.

Suppose that w ∈ ΛkV. Then w is a multilinear mapping of V* to K, so it is defined by its values on the k-fold Cartesian productV* × V* × ... × V*. If u1, u2, ..., uk−1 are k − 1 elements of V*, then define

Suppose that V has finite dimension n. Then the interior product induces a canonical isomorphism of vector spaces

In the geometrical setting, a non-zero element of the top exterior power Λn(V) (which is a one-dimensional vector space) is sometimes called a volume form (or orientation form, although this term may sometimes lead to ambiguity.) Relative to a given volume form σ, the isomorphism is given explicitly by

If, in addition to a volume form, the vector space V is equipped with an inner product identifying V with V*, then the resulting isomorphism is called the Hodge dual (or more commonly the Hodge star operator)

The composite of ∗ with itself maps Λk(V) → Λk(V) and is always a scalar multiple of the identity map. In most applications, the volume form is compatible with the inner product in the sense that it is an exterior product of an orthonormal basis of V. In this case,

where I is the identity, and the inner product has metric signature (p,q) — p plusses and q minuses.

For V a finite-dimensional space, an inner product on V defines an isomorphism of V with V∗, and so also an isomorphism of ΛkV with (ΛkV)∗. The pairing between these two spaces also takes the form of an inner product. On decomposable k-vectors,

the determinant of the matrix of inner products. In the special case vi = wi, the inner product is the square norm of the k-vector, given by the determinant of the Gramian matrix (⟨vi, vj⟩). This is then extended bilinearly (or sesquilinearly in the complex case) to a non-degenerate inner product on ΛkV. If ei, i=1,2,...,n, form an orthonormal basis of V, then the vectors of the form

Suppose that V and W are a pair of vector spaces and f : V → W is a linear transformation. Then, by the universal construction, there exists a unique homomorphism of graded algebras

such that

In particular, Λ(f) preserves homogeneous degree. The k-graded components of Λ(f) are given on decomposable elements by

Let

The components of the transformation Λ(k) relative to a basis of V and W is the matrix of k × k minors of f. In particular, if V = W and V is of finite dimension n, then Λn(f) is a mapping of a one-dimensional vector space Λn to itself, and is therefore given by a scalar: the determinant of f.

If K is a field of characteristic 0,[18] then the exterior algebra of a vector space V can be canonically identified with the vector subspace of T(V) consisting of antisymmetric tensors. Recall that the exterior algebra is the quotient of T(V) by the ideal I generated by x ⊗ x.

Let Tr(V) be the space of homogeneous tensors of degree r. This is spanned by decomposable tensors

The antisymmetrization (or sometimes the skew-symmetrization) of a decomposable tensor is defined by

where the sum is taken over the symmetric group of permutations on the symbols {1,...,r}. This extends by linearity and homogeneity to an operation, also denoted by Alt, on the full tensor algebra T(V). The image Alt(T(V)) is the alternating tensor algebra, denoted A(V). This is a vector subspace of T(V), and it inherits the structure of a graded vector space from that on T(V). It carries an associative graded product defined by

Although this product differs from the tensor product, the kernel of Alt is precisely the ideal I (again, assuming that K has characteristic 0), and there is a canonical isomorphism

In applications to linear algebra, the exterior product provides an abstract algebraic manner for describing the determinant and the minors of a matrix. For instance, it is well known that the magnitude of the determinant of a square matrix is equal to the volume of the parallelotope whose sides are the columns of the matrix. This suggests that the determinant can be defined in terms of the exterior product of the column vectors. Likewise, the k×k minors of a matrix can be defined by looking at the exterior products of column vectors chosen k at a time. These ideas can be extended not just to matrices but to linear transformations as well: the magnitude of the determinant of a linear transformation is the factor by which it scales the volume of any given reference parallelotope. So the determinant of a linear transformation can be defined in terms of what the transformation does to the top exterior power. The action of a transformation on the lesser exterior powers gives a basis-independent way to talk about the minors of the transformation.

The decomposable k-vectors have geometric interpretations: the bivector u ∧ v represents the plane spanned by the vectors, "weighted" with a number, given by the area of the oriented parallelogram with sides u and v. Analogously, the 3-vector u ∧ v ∧ w represents the spanned 3-space weighted by the volume of the oriented parallelepiped with edges u, v, and w.

The exterior algebra has notable applications in differential geometry, where it is used to define differential forms. A differential form at a point of a differentiable manifold is an alternating multilinear form on the tangent space at the point. Equivalently, a differential form of degree k is a linear functional on the k-th exterior power of the tangent space. As a consequence, the exterior product of multilinear forms defines a natural exterior product for differential forms. Differential forms play a major role in diverse areas of differential geometry.

Let L be a Lie algebra over a field K, then it is possible to define the structure of a chain complex on the exterior algebra of L. This is a K-linear mapping

defined on decomposable elements by

The Jacobi identity holds if and only if ∂∂ = 0, and so this is a necessary and sufficient condition for an anticommutative nonassociative algebra L to be a Lie algebra. Moreover, in that case ΛL is a chain complex with boundary operator ∂. The homology associated to this complex is the Lie algebra homology.

The exterior algebra was first introduced by Hermann Grassmann in 1844 under the blanket term of Ausdehnungslehre, or Theory of Extension.[19] This referred more generally to an algebraic (or axiomatic) theory of extended quantities and was one of the early precursors to the modern notion of a vector space. Saint-Venant also published similar ideas of exterior calculus for which he claimed priority over Grassmann.[20]

The algebra itself was built from a set of rules, or axioms, capturing the formal aspects of Cayley and Sylvester's theory of multivectors. It was thus a calculus, much like the propositional calculus, except focused exclusively on the task of formal reasoning in geometrical terms.[21] In particular, this new development allowed for an axiomatic characterization of dimension, a property that had previously only been examined from the coordinate point of view.

The import of this new theory of vectors and multivectors was lost to mid 19th century mathematicians,[22] until being thoroughly vetted by Giuseppe Peano in 1888. Peano's work also remained somewhat obscure until the turn of the century, when the subject was unified by members of the French geometry school (notably Henri Poincaré, Élie Cartan, and Gaston Darboux) who applied Grassmann's ideas to the calculus of differential forms.

A short while later, Alfred North Whitehead, borrowing from the ideas of Peano and Grassmann, introduced his universal algebra. This then paved the way for the 20th century developments of abstract algebra by placing the axiomatic notion of an algebraic system on a firm logical footing.

^Strictly speaking, the magnitude depends on some additional structure, namely that the vectors be in a Euclidean space. We do not generally assume that this structure is available, except where it is helpful to develop intuition on the subject.

^Grassmann (1844) introduced these as extended algebras (cf. Clifford 1878). He used the word äußere (literally translated as outer, or exterior) only to indicate the produkt he defined, which is nowadays conventionally called exterior product, probably to distinguish it from the outer product as defined in modern linear algebra.

^The term k-vector is not equivalent to and should not be confused with similar terms such as 4-vector, which in a different context could mean a 4-dimensional vector. A minority of authors use the term k-multivector instead of k-vector, which avoids this confusion.

^This part of the statement also holds in greater generality if V and W are modules over a commutative ring: That Λ converts epimorphisms to epimorphisms. See Bourbaki (1989, Proposition 3, III.7.2).

^This statement generalizes only to the case where V and W are projective modules over a commutative ring. Otherwise, it is generally not the case that Λ converts monomorphisms to monomorphisms. See Bourbaki (1989, Corollary to Proposition 12, III.7.9).

^Such a filtration also holds for vector bundles, and projective modules over a commutative ring. This is thus more general than the result quoted above for direct sums, since not every short exact sequence splits in other abelian categories.

This is the main mathematical reference for the article. It introduces the exterior algebra of a module over a commutative ring (although this article specializes primarily to the case when the ring is a field), including a discussion of the universal property, functoriality, duality, and the bialgebra structure. See chapters III.7 and III.11.

Includes applications of the exterior algebra to differential forms, specifically focused on integration and Stokes's theorem. The notation ΛkV in this text is used to mean the space of alternating k-forms on V; i.e., for Spivak ΛkV is what this article would call ΛkV*. Spivak discusses this in Addendum 4.